Note by C.M.: After talking with my friend Yuri Lima (a 3rd year PhD student at IMPA, currently at Columbus, Ohio, working with Vitaly Bergelson), I proposed to him to write some posts for this blog about the topics of his interest. He accepted my invitation and started a post (see below) containing an overview of his plans. Enjoy it!
We begin with a question: what conditions a set must have to possess arbitrarily long arithmetic progressions? Well, if this set is very sparse (such as the powers of ), there is no chance for such thing. On the other hand, a set with arbitrarily large intervals trivially satisfies it. Althought the precise condition is not known, there is one of great interest which is sufficient. Define the density of as
(Here, stands for the cardinality of the set ). Such limit not always exists, so that it is more convenient to consider the upper density of :
This is a well-defined number between and . In , Erdös and Turán conjectured that if , then has arbitrarily long arithmetic progressions. It remained wide open until , when Roth proved that such sets contain progression of lenght three. Later, Szemerédi, in , proved that they also have progressions of lenght four and, finally, in he solved the conjecture.
Theorem (Szemerédi, ). If has positive upper density, then it contains arbitrarily long arithmetic progressions.
His proof is a very hard combinatorial argument and relies in the Szemerédi’s Regularity Lemma (which we intend to talk in the future).
Breakthrough and the birth of a new area.
Two years later, Hillel Furstenberg gave another proof of Szemerédi’s Theorem, based on an deep analysis of the structure of general measure-preserving systems, known as Furstenberg’s Structural Theorem (see this lecture of Terence Tao for a discussion of this result in the case of distal systems). This gave birth to a new area, called Ergodic Ramsey Theory. As the name suggests, Ergodic Ramsey Theory deals with the use of Ergodic Theory (and related areas, such as topological dynamics) machinery to prove Ramsey Theory (and related combinatorial) problems.
In the next posts, we plan to discuss this interaction. Here is a sketch:
3. Polynomial Von Neumann’s Theorem.
7. Topological Dynamics and Van der Waerden’s Theorem.
8. Two simple models of measure-preserving systems: compact and weak mixing systems.
10. A glance at Furstenberg’s Structural Theorem and the proof of Multiple Poincaré’s Recurrence Theorem.
11. Generalized ergodic avergares: and a.e. convergence.
12. Green-Tao’s Theorem on the existence of arbitrarily long arithmetic progressions of primes.
The posts will be tagged by ERT+(number of the lecture).